Common fixed point theorems for nonlinear contractive mappings in fuzzy metric spaces
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RESEARCH
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Common fixed point theorems for nonlinear contractive mappings in fuzzy metric spaces ´ c3 Shenghua Wang1* , Saud M Alsulami2 and Ljubomir Ciri´ *
Correspondence: [email protected] 1 Department of Mathematics and Physics, North China Electric Power University, Baoding, 071003, China Full list of author information is available at the end of the article
Abstract In this paper, we prove several common fixed point theorems for nonlinear mappings with a function φ in fuzzy metric spaces. In these fixed point theorems, very simple conditions are imposed on the function φ . Our results improve some recent ones in the literature. Finally, an example is presented to illustrate the main result of this paper. MSC: 54E70; 47H25 Keywords: fuzzy metric space; contraction; Cauchy sequence; fixed point theorem
1 Introduction The concept of fuzzy metric spaces was defined in different ways [–]. Grabiec [] presented a fuzzy version of the Banach contraction principle in a fuzzy metric space in Kramosi and Michalek’s sense. Fang [] proved some fixed point theorems in fuzzy metric spaces, which improved, generalized, unified and extended some main results of Edelstein [], Istratescu [], Sehgal and Bharucha-Reid []. In order to obtain a Hausdorff topology, George and Veeramani [, ] modified the concept of fuzzy metric space due to Kramosil and Michalek []. Many fixed point theorems in complete fuzzy metric spaces in the sense of George and Veeramani (GV) [, ] have been obtained. For example, Singh and Chauhan [] proved some common fixed point theorems for four mappings in GV fuzzy metric spaces. Gregori and Sapena [] proved that each fuzzy contractive mapping has a unique fixed point in a complete GV fuzzy metric space, in which fuzzy contractive sequences are Cauchy. In , Bhaskar and Lakshmikantham [] introduced the concept of coupled fixed point in metric spaces and obtained some coupled fixed point theorems with the application to a bounded value problem. Based on Bhaskar and Lakshmikantham’s work, many researchers have obtained more coupled fixed point theorems in metric spaces and cone metric spaces; see [, ]. Recently, the investigation of coupled fixed point theorems has been extended from metric spaces to probabilistic metric spaces and fuzzy metric spaces; see [–]. In [], the authors gave the following results. Theorem SAS [, Theorem .] Let a ∗ b > ab for all a, b ∈ [, ] and let (X, M, ∗) be a complete fuzzy metric space such that M has an n-property. Let F : X × X → X and g : X → X be two functions such that M F(x, y), F(u, v), kt ≥ M(gx, gu, t) ∗ M(gy, gv, t) © 2013 Wang et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Wang et al. Fixed Point Theory and Applications 2013, 2013:191 http://www.fixedpointtheoryandapplica
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